{
  "id": "2102.06264",
  "version": "v1",
  "published": "2021-02-11T20:41:07.000Z",
  "updated": "2021-02-11T20:41:07.000Z",
  "title": "On The Gersten-Witt Complex of an Azumaya Algebra with Involution",
  "authors": [
    "Uriya A. First"
  ],
  "comment": "19 pages. Comments are welcome. This is used to be the appendix of arXiv:1911.07666v1 (which was removed in arXiv:1911.07666v2)",
  "categories": [
    "math.AG",
    "math.KT",
    "math.NT"
  ],
  "abstract": "Let $(A,\\sigma)$ be an Azumaya algebra with involution over a regular ring $R$. We prove that the Gersten-Witt complex of $(A,\\sigma)$ defined by Gille is isomorphic to the Gersten-Witt complex of $(A,\\sigma)$ defined by Bayer-Fluckiger, Parimala and the author. Advantages of both constructions are used to show that the Gersten-Witt complex is exact when $\\dim R\\leq 3$, $\\mathrm{ind}\\, A\\leq 2$ and $\\sigma$ is orthogonal or symplectic. This means that the Grothendieck-Serre conjecture holds for the group $R$-scheme of $\\sigma$-unitary elements in $A$ under the same hypotheses; $R$ is not required to contain a field.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-02-11T20:41:07.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11E57",
      "11E39",
      "16H05",
      "19G38"
    ],
    "keywords": [
      "gersten-witt complex",
      "azumaya algebra",
      "involution",
      "grothendieck-serre conjecture holds",
      "unitary elements"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}